Jacob (or James, or Jacques), the son of a
prosperous Protestant merchant, was the first of the remarkable Bernoulli
mathematical dynasty. Back in 1583, the Bernoulli family had fled from Antwerp,
first to Frankfort and then to Basel, to escape the persecutions of the
Catholic Duke of Alba, acting for the absentee King of Spain. Jacob was
first trained in theology, but after receiving his degree and working as
a tutor in Geneva for two years, he made his own escape into mathematics.
He went to France, where he studied with followers of Descartes. He visited
the Netherlands, and then went to England, where he met Boyle and Hooke.
On his return to Switzerland in 1683 he began teaching mechanics at
the University of Basel. He published papers on algebra and probability
in 1685 and on geometry in 1687, and was appointed to the chair of mathematics
at Basel in 1687. At about this time his younger brother Johannes, who had
been studying mechanics, asked Jacob to teach him mathematics, beginning
with Leibniz's difficult 1684 paper on the calculus, the big new thing at
that time. The two collaborated for a while, but a period of rivalry and
mutual attacks followed, becoming even more acrimonious after Johannes moved
to the Netherlands, in 1697.

Jacob's contributions to statistics include
a major 1689 work on infinite series, in which his statement of the Law
of Large Numbers appeared. The essence of the law is that, given an inherent
probability that an event will come out a certain way, an increasing number
of trials will give an increasingly accurate approximation to that frequency,
random variations on either side of the inherently probable value tending
to cancel out in the long run. Jacob's work on the calculus (he was the
first to use the term "integral") and on geometry gave him a deep insight
into the equations of many of the curves that are useful in describing growth
and probability. The curve called the Lemniscate of Bernoulli still bears
his name, as does the Bernoulli equation y' = p(x)y + q(x)y*n. He was so
taken with another curve he investigated, the logarithmic (equilangular)
spiral, that he ordered it to be engraved on his tomb with the caption Eadem
mutata resurgo: "Though transformed, I shall arise unchanged." Thus
did mathematics and theology join hands at the end. After Jacob's death,
Johannes realized his long ambition by succeeding to his brother's chair
at Basel.

Jacob's most important work was the Ars Conjectandi,
which was published only after his death (in 1713). With de
Moivre's work of 1718, it is the first major treatise in the field of
probability and statistics. Bernoulli's legacy, like his life, has an oppositional
character. The Ars Conjectandi defines a "frequentist" or objective
position as against the "expectation" or subjective position which has developed
from the work of Bayes. Opposition between these two viewpoints still to
some extent divides statistics into two hostile camps. One subgroup of the
International Association for Statistics is called the Bernoulli Society.

Bernoulli developed the binomial approach of
Pascal, in which the binomial
coefficients of the Arithmetical Triangle have a central place. A coin
toss or other trial which can have only one of two outcomes is sometimes
called a Bernoulli trial. The world owes Bernoulli no lesser tribute.